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Statistical inversion method for binary asteroids orbit determination
Irina Kovalenko„ Daniel Hestroffer, A. Doressoundiram, Nicolai Emelyanov, Radu Stoica
To cite this version:
Irina Kovalenko„ Daniel Hestroffer, A. Doressoundiram, Nicolai Emelyanov, Radu Stoica. Statistical inversion method for binary asteroids orbit determination. Journées systèmes de référence spatio-temporels, Sep 2014, St Petersburg, Russia. �hal-01103904�
Introduction
We focus on the study of binary asteroids, which are common in the Solar system from its inner to its outer regions. These objects provide fundamental physical parameters such as mass and density, and hence clues on the early Solar System, or other processes that are affecting asteroid over time. The present method of orbit computation for resolved binaries is based on Markov Chain Monte-Carlo statistical inversion technique. Particularly, we use the Metropolis-Hasting algorithm with Thiele-Innes equation for sampling the orbital elements and system mass through the sampling of observations. The method requires a minimum of four observations, made at the same tangent plane; it is of particular interest for orbit determination over short arcs or with sparse data. The observations are sampled within their observational errors with an assumed distribution. The sampling yields the whole region of possible orbits, including the one that is most probable.
Statistical inversion method for binary
asteroids orbit determination
I.Kovalenko, D. Hestroffer, A. Doressoundiram, N. Emelyanov, R. Stoica [email protected] (IMCCE - LESIA, Paris Observatory)
Orbit determination
References
1) R. Palacios 1958 AJ 63, 395 2) D. Oszkiewicz et al. 2013 ,SF2A 237
The astrometric observations are related to
the theoretical positions through the
observational equation:
𝝋 = 𝜓 𝑿 + 𝜀
• Observations: 𝝋 = 𝜌1, 𝜃1; … ; 𝜌N, 𝜃N
• Sky-plane position: 𝜓 𝑿
• Orbital elements + system’s mass: 𝑿 = (𝑎, 𝑒, 𝑖 , Ω, 𝜔, 𝑀, 𝑚𝑠𝑦𝑠)
• Observational errors: 𝜀 = (𝜀 𝛼1, 𝜀𝛿1; … ; 𝜀 𝛼𝑁, 𝜀𝛿𝑁)
𝑝 𝑋 𝜑 =
𝑝 𝜑 𝑋 𝑝 𝑋
𝑝(𝜑)
Markov Chain Monte-Carlo method
𝑝 𝑋 𝜑 ∝ 𝑝 𝜑 𝑋 𝑝 𝑋
Where • 𝑝(𝑋) ∝ 𝑑𝑒𝑡Λ−1 • 𝑝 𝝋 𝑿 = 𝑝𝜀(observational error p. d. f. ) = exp[− 1 2 𝝋 − 𝜓 𝑿 𝑇 Λ−1 𝝋 − 𝜓 𝑿 ]N observations
𝑺 = 𝜌1, 𝜃1; 𝜌2, 𝜃2; 𝜌3, 𝜃3; 𝜌4 Select a random set of 4observations
The Metropolis–Hastings algorithm will be used for sampling the parameters 𝑿.
For each t iteration:
𝑺′- proposal set of observations
𝑺𝑡- last accepted set of observations
p𝑡(𝑺′; 𝑆𝑡)- proposal density
𝛼𝑖′ ∝ 𝐺 𝛼𝑖𝑡, 𝜎(𝛼𝑖) , 𝛿𝑖′ ∝ 𝐺 𝛿𝑖𝑡, 𝜎(𝛿𝑖)
Orbit 𝑿′+ monitor the fit to all observations
𝑎 = 𝑝 𝑋
′|𝜑 |𝐽𝑡|
𝑝 𝑋t|𝜑 |𝐽′| 𝐴𝑐𝑐𝑒𝑝𝑡𝑒𝑛𝑐𝑒 𝑐𝑟𝑖𝑡𝑒𝑟𝑖𝑎
[2] 𝐽′ and 𝐽𝑡 - the determinants of Jacobians from coordinates to
orbital parameters 𝐽 = det 𝜕𝑆 𝜕𝑋
𝑿𝑡+1 = 𝑿′ 𝑿𝒕+𝟏 = 𝑿′, with probability 𝑎
or 𝑿𝒕+𝟏 = 𝑿𝒕 ,with probability 1 − 𝑎
Thiele – Innes method [1]:
𝑡𝑖 − 𝑡𝑗 − Δ𝑖𝑗 𝑐 = 1 𝜇 [ 𝐸𝑖 − 𝐸𝑗 − sin (𝐸𝑖 − 𝐸𝑗)] 𝑖, 𝑗 = 1, 2, 3,4 Δ𝑖𝑗 = 𝜌𝑖𝜌𝑗sin (𝜃𝑗 − 𝜃𝑖) => 𝑿 = (𝑎, 𝑒 𝑖 , 𝜔, Ω, 𝑀, 𝑇) 𝐼𝑓 𝑎 ≥ 1 𝐼𝑓 𝑎 < 1
This process is repeated until the stationary a posteriori density is reached.
z, km
x, km y, km
Acknowledgements
This work is supported by the Labex ESEP (ANR N° 2011-LABX-030).
The algorithm is run for a large number of iterations until the entire possible orbital-element space is mapped
